Finite Group Databasebeta

← #16821140

Subgroups — #16821140

Order 3712.

266 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14271728C211yesmaximal, characteristic
2#14271144C211yesmaximal, characteristic, complemented
3#14271135C211yesmaximal, characteristic, complemented
4#14270951C211yesmaximal, characteristic, complemented
5#14270950C211yesmaximal, characteristic, complemented
6#14270949C211yesmaximal, characteristic, complemented
7C29×AΣL(1,𝔽16)C211yesmaximal, characteristic, complemented
8C22×C22⋊D29C411yescharacteristic, complemented
9C2×#90807C2211yescharacteristic
10C29×#120C2211yescharacteristic
11#11745315C411yescharacteristic, complemented
12#11745302—21no—
13#11745301—21no—
14#11745299C2211yescharacteristic
15M16×D29—21no—
16#11745247—21no—
17#11745246—21no—
18#11745244C411yescharacteristic, complemented
19#11745239—21no—
20#11745239C411yescharacteristic, complemented
21C29×#100C2211yescharacteristic
22C29×#99C2211yescharacteristic
23#11745182C2211yescharacteristic
24#11745181C2211yescharacteristic
25#11745172—21no—
26#11745154—21no—
27C24×C29C2×C411yescharacteristic
28C23×D29C2×C411yescharacteristic
29C2×D4×C29—21no—
30C2×D4×C29C2311yescharacteristic
31C2×C22⋊D29—21no—
32C2×C22⋊D29—21no—
33C2×C22⋊D29—21no—
34C2×C22⋊D29—21no—
35C2×C22⋊D29—41no—
36C2×C22⋊D29—41no—
37C2×C22⋊D29C2×C411yescharacteristic
38C2×C22⋊D29D411yescharacteristic
39C22×Dic29C2×C411yescharacteristic
40C22×Dic29D411yescharacteristic
41#90807—41no—
42#90807—41no—
43C2×D116—21no—
44C2×C4×D29D411yescharacteristic
45C2×Dic58D411yescharacteristic
46M16×C29—21no—
47C22⋊C4×C29—21no—
48C22⋊C4×C29—21no—
49C22⋊C4×C29C2×C411yescharacteristic
50C22⋊Dic29—21no—
51C22⋊Dic29—21no—
52C22⋊Dic29C2×C411yescharacteristic
53#90781—21no—
54#90781—21no—
55#90779—21no—
56#90779—41no—
57C4×Dic29—21no—
58C4×Dic29—41no—
59#90777—21no—
60#90772—41no—
61C8×D29—41no—
62C23×C29—21no—
63C23×C29—41no—
64C23×C29—81no—
65C23×C29C22×C411yescharacteristic
66C23×C29C2×D411yescharacteristic
67C22×D29—21no—
68C22×D29—41no—
69C22×D29—81no—
70C22×D29C22⋊C411yescharacteristic
71D4×C29—41no—
72D4×C29—81no—
73C2×C4×C29—21no—
74C2×C4×C29—41no—
75C2×C4×C29C2×D411yescharacteristic
76C22⋊D29—41no—
77C22⋊D29—41no—
78C22⋊D29—42no—
79C22⋊D29—41no—
80C22⋊D29—41no—
81C22⋊D29—81no—
82C22⋊D29—81no—
83C2×Dic29—21no—
84C2×Dic29—21no—
85C2×Dic29—21no—
86C2×Dic29—21no—
87C2×Dic29—21no—
88C2×Dic29—41no—
89C2×Dic29C22⋊C411yescharacteristic
90C2×Dic29C22⋊C411yescharacteristic
91C2×Dic29C22⋊C411yescharacteristic
92D116—81no—
93C4×D29—21no—
94C4×D29—21no—
95Dic58—41no—
96C8×C29—41no—
97#6597—41no—
98#2122—291nomaximal, Sylow
99C22×C29—21no—
100C22×C29—21no—
101C22×C29—41no—
102C22×C29—41no—
103C22×C29—41no—
104C22×C29—41no—
105C22×C29—41no—
106C22×C29—82no—
107C22×C29C2×C22⋊C411yescharacteristic
108C2×D29—21no—
109C2×D29—41no—
110C2×D29—41no—
111C2×D29—41no—
112C2×D29—41no—
113C2×D29—82no—
114C4×C29—21no—
115C4×C29—41no—
116C4×C29—81no—
117Dic29—21no—
118Dic29—21no—
119Dic29—41no—
120Dic29—81no—
121Dic29#9911yescharacteristic
122Dic29#9911yescharacteristic
123#535—291no—
124#413—291no—
125#410—291no—
126AΣL(1,𝔽16)—291no—
127AΣL(1,𝔽16)—291no—
128AΣL(1,𝔽16)—291no—
129AΣL(1,𝔽16)D2911yescharacteristic, complemented
130C2×C29—21no—
131C2×C29—41no—
132C2×C29—41no—
133C2×C29—81no—
134C2×C29C2×#9911yescharacteristic
135D29—41no—
136D29—81no—
137C2×#41—291no—
138C22×D4—291no—
139C2×M16—581no—
140#127—581no—
141#124—581no—
142#121—581no—
143#121—581no—
144#120—291no—
145#120—291no—
146#120—291no—
147#120C2×D2911yescharacteristic
148#117—291no—
149#117—581no—
150#101—581no—
151#100—291no—
152#100C2×D2911yescharacteristic
153#99—291no—
154#99—581no—
155#99C2×D2911yescharacteristic
156C29#212211yescharacteristic, complemented, Sylow
157C24—291no—
158C24C4×D2911yescharacteristic
159#41—1161no—
160#41—1161no—
161C2×Q8—291no—
162C2×D4—21no—
163C2×D4—291no—
164C2×D4—291no—
165C2×D4—581no—
166C2×D4—581no—
167C2×D4—581no—
168C2×D4—581no—
169C2×D4—581no—
170C2×D4—1161no—
171C2×D4—1161no—
172C2×D4C22×D2911yescharacteristic
173C22×C4—291no—
174C22×C4—291no—
175C22×C4—291no—
176M16—21no—
177M16—581no—
178M16—1161no—
179C2×C8—1161no—
180C4⋊C4—581no—
181C4⋊C4—1161no—
182C22⋊C4—21no—
183C22⋊C4—21no—
184C22⋊C4—291no—
185C22⋊C4—581no—
186C22⋊C4—581no—
187C22⋊C4—581no—
188C22⋊C4—581no—
189C22⋊C4C4×D2911yescharacteristic
190C42—581no—
191C42—1161no—
192C23—21no—
193C23—41no—
194C23—81no—
195C23—291no—
196C23—581no—
197C23—1161no—
198C23—2321no—
199C23C2×C4×D2911yescharacteristic
200C23D4×D2911yescharacteristic
201Q8—1161no—
202D4—41no—
203D4—81no—
204D4—1161no—
205D4—1161no—
206D4—1161no—
207D4—1161no—
208D4—1162no—
209D4—2321no—
210D4—2321no—
211D4—2321no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—291no—
215C2×C4—291no—
216C2×C4—291no—
217C2×C4—581no—
218C2×C4—581no—
219C2×C4—581no—
220C2×C4—581no—
221C2×C4—581no—
222C2×C4—581no—
223C2×C4—581no—
224C2×C4—1161no—
225C2×C4D4×D2911yescharacteristic
226C8—41no—
227C8—1161no—
228C22—21no—
229C22—21no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22—82no—
236C22—581no—
237C22—1161no—
238C22—1161no—
239C22—1161no—
240C22—1161no—
241C22—2322no—
242C22C22⋊C4×D2911yescharacteristic
243C4—21no—
244C4—41no—
245C4—81no—
246C4—291no—
247C4—291no—
248C4—581no—
249C4—581no—
250C4—1161no—
251C4—2321no—
252C2—21no—
253C2—41no—
254C2—41no—
255C2—81no—
256C2—1161no—
257C2—2321no—
258C2#99×D2911yescharacteristic
subgroupG/H
centerC2#99×D29
commutatorC23×C29C22×C4
FrattiniC2×D4C22×D29
FittingC29×AΣL(1,𝔽16)C2
socleC2×C29C2×#99
prime radical#14271728C2
cosocleC2×D4×C29C23
hypercenterAΣL(1,𝔽16)D29
hypocenterC29#2122
LCSG ▷ C23×C29 ▷ C22×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ AΣL(1,𝔽16)

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #14271144C2semidirect
3. #14271135C2semidirect
4. #14270951C2semidirect
5. #14270950C2semidirect
6. #14270949C2semidirect
7. C29×AΣL(1,𝔽16)C2semidirect
8. C22×C22⋊D29C4semidirect
11. #11745315C4semidirect
18. #11745244C4semidirect
20. #11745239C4semidirect
129. AΣL(1,𝔽16)D29semidirect
156. C29#2122semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#142717282
#142711442
#142711352
#142709512
#142709502
#142709492
C29×AΣL(1,𝔽16)2
#212229